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PONTTFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28

where A, B, and C are matrices A,, B,, and C, evaluated at

pu=p; (i=0, 1, ..., n). In view of (2), (9) and 3 g;=1 we find
i=1
that one of the characteristic numbers {4,, …, |, is I+r. We
can also show that the eigen-vector % associated with 1 +7 is
non-negative. It is clear that we have

(16) x B(1+—d)+(1+n)x([—+A—B—C)=o.

Thus (14) has a particular solution (1 + 7)! which is referred to
as a balanced growth solution. We also refer to a state fulfilling
 (8) (9), and (16) as a state of balanced growth.

2. So far we have treated the rate of interest as a given
constant and have shown that to any assigned value of it there
corresponds a state of balanced growth. It is impossible, however,
 for the rate of growth of outputs to exceed the rate of
growth of the working population for a long time, because the
scarcity of labour will sooner or later emerge. In the contrary
case where the labour force is increasing at a rate higher than
the rate of growth of outputs, the ratio of the number of unemployed
 to the number of employed workers continues to rise.
In the following, therefore, we are concerned with finding a
rate of balanced growth at which the growth of outputs is in
harmony with that of the labour force.
We begin with examining the effects of a change in the
rate of interest on the long-run equilibrium prices. Differentiating
 (10) with respect to », we get

a log :
ax — (us — @-— 0)"

ar

y| Morishima - pag. 8